We build limit spaces
of manifolds
with uniform lower bounds on Ricci curvature such that
is nowhere a topological manifold, and in fact every open set
has
infinitely generated homology.
More completely, it is known that any such
must be
-rectifiable for some
unique
. It is also
known that if
, then
is a topological
manifold on an open dense subset, and it has been an open question as to whether this holds for
. Consider now any
smooth complete
-manifold
with
and
. Then for each
we construct a
complete
-rectifiable
metric space
with
such that the
following hold. First,
is a limit space
,
where
are smooth manifolds satisfying the same lower Ricci bound
.
Additionally,
has no open subset which is topologically a manifold. Indeed, for
any open
we have that the second
homology
is infinitely
generated. Topologically,
is the connect sum of
with an infinite number of densely spaced copies of
.
In this way we see that every
-manifold
may be approximated arbitrarily
closely by
-dimensional
limit spaces
which are nowhere manifolds. We will see that there is a sense, as yet imprecise, in
which generically one should expect manifold structures to not exist on spaces with
higher-dimensional Ricci curvature lower bounds.
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